Acoustic and electromagnetic scattering analysis using discrete sources II the scalar helmholtz equation

Acoustic and electromagnetic scattering analysis using discrete sources II the scalar helmholtz equation Acoustic and electromagnetic scattering analysis using discrete sources II the scalar helmholtz equation Acoustic and electromagnetic scattering analysis using discrete sources II the scalar helmholtz equation Acoustic and electromagnetic scattering analysis using discrete sources II the scalar helmholtz equation Acoustic and electromagnetic scattering analysis using discrete sources II the scalar helmholtz equation

II THE SCALAR HELMHOLTZ EQUATION This chapter is devoted to presenting the foundations of obstacle scattering problems for tintie-harmonic acoustic waves We begin with a brief discussion of the physical background of the scattering problem, and then we will formulate the boundary-value problems for the Helmholtz equation We will synthetically recall the basic concepts as they were presented by Colton and Kress [32], [35] However, we decided to leave out some details in the analysis In this context we not repeat the technical proof for the jump relations and the regularity properties for single- and double-layer potentials with continuous densities Leaving aside these details, however, we will present a theorem given by Lax [90] which enables us to extend the jump relations from the case of continuous densities to square integrable densities We then establish some properties of surface potentials vanishing in sets of R^ These results play a significant role in our completeness analysis Discussing the Green representation theorems will enable us to derive some estimates of the solutions We will then analyze the general null-field equation for the exterior Dirichlet and Neumann problems In particular, we will establish the existence and uniqueness of the solutions and will prove the equivalence of the null-field equations with some boundary integral equations 17 18 CHAPTER II THE SCALAR HELMHOLTZ EQUATION BOUNDARYVALUE PROBLEMS IN ACOUSTIC THEORY Acoustic waves are associated only with local motions of the particles of the fluid and not with bodily motion of the fluid itself The field variables of interest in a fluid are the particle velocity v' = v'(x,^), pressure p' = p'(x,t), mass density p' = p\x.^t) and the specific entropy ' = 5'(x,t) To derive the diff^erential equations describing acoustic fields we assume that each of these variables undergoes small fluctuations about their mean values: Vo = 0, P05 Po and SQ Generally, quadratic terms in particle velocity, pressure, density and entropy fluctuations are neglected and conservation laws for mass and momentum are linearized in terms of the fluctuations V = v(x,^), p = p(x,t), p = p{x^t) and S = 5(x,t) In this context the motion is governed by the linearized Euler equation dv ^ + - V p = 0, (2.1) and the linearized equation of continuity | ^ + PoV-v = (2.2) From thermodinamics we can write the pressure as a function of density and entropy If we assume that the acoustic wave propagation is an adiabatic process at constant entropy and the changes in density are small, we have the linearized state equation ^ = g;^(Po,5o)^ (2.3) Defining the speed of acoustic waves via c^ = f^{p„So) (2.4) we see that the pressure satisfies the time-dependent wave equation Taking the curl of the linearized Euler equation we get V X V= (2.6) V = —V[/, Po (2.7) and therefore we can take BOUNDARY-VALUE PROBLEMS IN ACOUSTICS 19 where f/ is a scalar field called the velocity potential We mention that the above equation is a direct consequence of the assumption of a nonviscous fluid Further, substituting (2.7) in (2.1) we obtain and clearly the velocity potential also satisfies the time-dependent wave equation :^-^ = ^u (2.9) For time-harmonic acoustic waves of the form U{x,t) = Re {i/(x)e-^^*} (2.10) with frequency a; > 0, we deduce that (2.9) can be transformed to the well-known reduced wave equation or Helmholtz equation Au-j-k^u^O, (2.11) where the wave number k is given by the positive constant k = u/c If we consider the acoustic wave propagation in a medium with damping coefficients C» then the wave number is given byfc^= a; (a; + jQ /c^- We choose the sign of k such that Imfc> Before we consider the boundary-value problems for the Helmholtz equation let us introduce some normed spaces which are relevant for acoustic scattering Let G be a closed subset of R^ By C{G) we denote the linear space of all continuous complex-valued functions defined on G C{G) is a Banach space equipped with the supremum norm ll«llcx),G =S^P I^WIx€G By L^{G) we denote the Hilbert space of all square integrable functions on G, i.e L'^{G) ' dG exists = < a / a : G -^ C, a measurable, / \a\ I G L^{G) is the completion of C{G) with respect to the square-integral norm IHl2,G=(/N'dG 20 CHAPTER II THE SCALAR HELMHOLTZ EQUATION induced by the scalar product G The H5lder space or the space of uniformly Holder continuous functions C^'"(G) is the linear space of all complex-valued functions defined on G which are bounded and uniformly Holder continuous with exponent a A function a : G —> C is called uniformly Holder continuous with Holder exponent < a < if Kx)-a(y)| We note here that one can pose and solve the boundary-value problems for the boundary conditions in an L^-sense The existence results are then obtained under weaker regularity assumptions on the given boundary data On the other hand, the assumption on the boundary to be of class C^ is connected with the integral equation approach which is used to prove the existence of solutions for scattering problems Actually it is possible to allow Lyapunov boundaries instead of C^ boundaries and still have compact operators The situation changes considerable for Lipschitz domains Allowing such nonsmooth domains and 'rough' boundary data drastically changes the nature of the problem since it affects the compactness of the boundary integral operators In fact, even proving the very boundedness of these operators becomes a fundamentally harder problem A basic idea, going back to Rellich [130] is to use the quantitative version of some appropriate integral identities to overcome the lack of compactness of the boundary integral operators on Lipschitz boundaries For more details we refer to Brown [17] and Dahlberg et al [37] SINGLE- AND DOUBLE-LAYER POTENTIALS We briefly review the basic jump relations and regularity properties for acoustic single- and double-layer potentials 24 CHAPTER II THE SCALAR HELMHOLTZ EQUATION Let be a surface of class C^ and let a be an integrable function Then Ua(x) =y^a(y)p(x,y,fc)d5(y), x G R^ - 5, (2.20) and t;„(x) = j a{y)^^^^dS{y), s x € R^ - 5, (2.21) are called the acoustic single-layer and acoustic double-layer potentials, respectively They satisfy the Helmholtz equation in Di and in Dg and the Sommerfeld radiation condition Here g is the Green function or the fundamental solution defined by ff(x,y,fc) = ^ ^ ^ ^ _ y ^ , x ^ y (2.22) The single-layer potential with continuous density a is uniformly Holder continuous throughout R^ and ||walL,R3 < Ca ||a|U,5 , < a < (2.23) For densities a G C^'"(S), < a < 1, the first derivatives of the singlelayer potential Ua can be uniformly extended in a Holder continuous fashion from Dg into Dg and from Di into Di with boundary values (Vt/a)± (x) = ja(y)Vx^(x,y,fc)d5(y) s T | a ( x ) n ( x ) , x € 5, (2.24) where {Vua)^ (x) = lim Vu(x ± ftn(x)) (2.25) in the sense of uniform convergence on S and where the integral exists as improper integral The same regularity property holds for the doublelayer potential Va with density a £ C^'^{S), < a < In addition, the first derivatives of the double-layer potential Va with density a C^'"(5), < a < 1, can be uniformly Holder continuously extended from Dg into Dg and from Di into Di The estimates IIVUalU.D < C a | | a | U , s , (2.26) \\VaL,D, < Ca ||a|L,s (2-27) SINGLE- AND DOUBLE-LAYER POTENTIALS 25 and (2.28) \\^VaL,Dr^Ca\\a\\,^^^S hold, where t stands for s and i In all inequalities the constant Ca depend on S and a For the single- and double-layer potentials with continuous density we have the following jump relations: (a) lim [ua (x ± hn{x)) - Ua{x)] = 0, - (c) / ' • ( ^ ' ^g(x,y,fc)^^, , _ = 0, lira = 0, Is I (d) lim ^ ( x + /in(x))-^(x-Mx)) = 0, (2.29) where x € and the integrals exist as improper integrals The single- and double layer operators and /C, and the normal derivative operator K.' will be frequently used in the sequel They are defined by (5a) (x) = jaiy)gix,y,k)dS{y),xeS, s (2.30) {Ka){x) = |o(y)^^^l^d5(y),xe5, and {IC'a) (x) = I a{y)^i^^dS{y), x (2.31) The operators 5, K and K' are compact in C{S) and C^'°^{S) for < a < 5, /C and /C' map C{S) into C^^'^iS), and and K map C^^'^iS) 26 CHAPTER II THE SCALAR HELMHOLTZ EQUATION into C^'^{S) We note that S is self adjoint and /C and /C' are adjoint with respect to the L^ biUnear form {a,b) = JabdS (2.32) (5a, b) = (a, 56) (2.33) {ICa.b) = {a.K'b) (2.34) that is and foralla,6GC(5) As shown by Kersten [79] the jump relations in L^(5) can be deduced from the classical results through the use of a functional analytic tool provided by Lax [90] This result can be stated as follows Let X\^ X2 be Banach spaces, and H\^ H2 be Hilbert spaces with continuous and dense embeddings X^ C i/t for i = 1,2 Let T : Xi-^ X2 and T^ : X2 -* Xi be two bounded linear operators such that T and T^ are adjoint to each other with respect to the scalar products of H\ and if2- Then T can be extended to a bounded operator from if into ii2 and \\nC(Hi,H2) < \\nC{Xt,Xj) r^ii C(X2,Xi) (2.35) We are now in position to formulate the following jump properties for the single- and double-layer potentials in terms of L^-continuity THEOREM 2.1: Let S be a closed surface of class C^ and let n denotes the unit outward normal to S Then for square integrable densities the behavior of surface potentials at the boundary is described by the following jump relations : (a) lim ||ua(.±/in(.))-tia(.)ll2,5 = 0' (b) lim h-.0+ dUg dn i.±hn{.)) - I „,„M^.,„„i„ is (c) lim Va{.±hn{.)) lim 2,5 - = 0, Is (d) = 0, 2,5 ^i.+hn{.))-^(.-hn(.)) = 2,S (2.36) SINGLE- AND DOUBLE-LAYER POTENTIALS 27 Proof: The proof of the theorem was given by Kersten [79] We outhne the proof for easy reference For a continuous density ao we rewrite the jump relations in compact form as lim ||Thao|U,5 = 0, (2.37) where (7^)o Since C{S) is dense in L'^{S) we find ao € C{S) such that \\a — aoHa ^ ^- ^^^ us choose /IQ such that | | ^ o o | | ^ < e for all h with < h < HQ, Then, < C||T^aom5-hM||a-ao||2,5 < (2.40) (C -f M)£ where the constants C and M not depend on e and /IQ Consequently, lim;i_^o+ 11^^112 = and the theorem is proved The analysis of the completeness of different systems of functions in L^(S) relies on the results of following theorems T H E O R E M 2.2: Consider Di a bounded domain of class C^ with boundary S Let the single-layer potential Ua with density a € L'^{S) satisfy Ua = in Di, (2.41) 28 CHAPTER II THE SCALAR HELMHOLTZ EQUATION Then a ~ on (a vanishes almost everywhere on S) Proof: From the jump relations for the normal derivative of the singlelayer potential with square integrable density lim dug •hn{.))- 2» + / « ( y ) d9i.,y,k) d5(y) an(.) = (2.42) 2,5 we find that the surface density a satisfies (2.43) almost everywhere on The above integral equation is a Predholm integral equation of the second kind The operator in the left-hand side of (2.43) is an elliptic pseudodifferential operator of order zero According to Mikhlin [101] we find that a - ao C{S) Since K' maps C{S) into C^^'^iS) we see that ao £ C^'"(5) Using the regularity results for the derivative of the single-layer potential we conclude that Uao belongs to C^'"(JDS) NOW the jump relations for the single-layer potential with continuous density show that Uao solves the homogeneous exterior Dirichlet problem Therefore Uao = in JD^) and hence, from duao^/On - duao^/dn «« ao ^ we get a '^ on The theorem is proved We mention that the equivalence a ^ ao € C^'^{S) can be obtained directly by using the following regularity result: if A is an elliptic pseudodifferential operator of order zero then any solution in C^'^{S) of the inhomogeneous equation Aa = / inherits additional smoothness from / , so that / G C'^^'^iS) implies that a C"''"(5), where m > and < a < 1; in particular, if a solves the homogeneous equation Aa = 0, then a e nm>oC^''*(S') C C ^ ( ) A similar result holds when Ua vanishes in the unbounded domain Dg T H E O R E M 2.3: Consider Di a bounded domain of class C^ with boundary S and exterior Dg, Assume k ^ p ( A ) O'l^d let the single-layer potential Ua with density a G L^{S) satisfy Ua = in Ds (2.44) Then a ^0 on S Proof: Repeating the arguments of the previous theorem we see that Uao y with a ~ ao G C^'°'{S)y solves the homogeneous interior Dirichlet problem The assumption k ^ p{Di) implies Uao = in A , and the proof can be completed as above For the double-layer potential we can state the following results T H E O R E M 2.4: Consider Dt a bounded domain of class C^ with boundary S Let the double-layer potential Va with density a G L^{S) satisfy Va =0 in Di, (2.45) 29 SINGLE- AND DOUBLE-LAYER POTENTIALS Then a^O on S Proof: The jump relation for the double-layer potential with square integrable density Um = Va(.-M-)) LS (2.46) 2,5 shows that the surface density a satisfies (i/-A:)« = (2.47) almost everywhere on S, Using the same arguments as in theorem 2.2 we obtain a ao e C{S) Then, since /C maps C{S) into C^^'^iS) and C^'"(5) into C^'^'iS) we deduce that ao E C^'"(5) Using the regularity results for the derivative of the double-layer potential we see that Vao belongs to C^'^{Ds)^ The jump relations for the normal derivative of the double-layer potential with continuous density shows that Vao solves the homogeneous exterior Neumann problem, and therefore Vao = in Ds Finally, from ^ao+ ~ '^ao- = ao = we get a '^ on T H E O R E M 2.5: Consider Di a bounded domain of class C^ with boundary S and exterior Ds- Assume k ^ 'n{Di) and let the double-layer potential Va with density a € L'^{S) satisfy Va = in Ds (2.48) Then a ^ on Proof: The proof proceeds as in theorem 2.4 Next we will consider combinations of single- and double-layer potentials T H E O R E M 2.6: Consider Di a bounded domain of class C^ with boundary S Let the combined potential Wa = Ua — Xva with density a € L'^{S) and Im{Xk) > satisfy (2.49) Wa = in Di Then a ~ on S, Proof: The idea of the proof is due to Hahner [71] The jump relations for the surface potentials with square integrable densities gives lim \\wa{.+hn{,)) + Aa|(2 c -0+ • h—^" lim dwg dn 0, (2.50) (.-f/in(.))-f a 2,S 30 CHAPTER II THE SCALAR HELMHOLTZ EQUATION Let us choose a parallel exterior surface 5^, Sh = { y / y = X -h /in(x), X G 5, /i > 0} Then, we have \f\a\^dS J s = = - lim /i->o+ J s lim fwa{.'^hn{.))a*{l'-2hH-hh'^K)dS fwa^dS, Sh (2.51) where H denotes the mean curvature and K the Gaussian curvature of S Let us now consider a spherical surface 5/? of radius R enclosing Di Application of the first Green theorem in the region DhR , bounded by the surface Sh and the spherical surface SR, yields IrnLfwa^ds) = lm(k J Wards' -Im(A;) f (\kf\waf + \Vwaf)dV (2.52) Letting /i —• 0+ and using (2.51) gives ImiXk) f\a\^dS s = Imlk \ fwa^dS ' SR ^^ (2.53) -lm{k) J {\kf\Wa\'' + \VWaf')dV, DR where ^lim^ j DhR {\k\^ \wa\^ + \Vwa\^) dV= f (l^l" l«^«l^ + IVwal') dV.- (2.54) DR 31 SINGLE- AND DOUBLE-LAYER POTENTIALS Then, taking into account the radiation condition (weak form) lim / {"^ - jkwal dS = lim SR / < SR W^KlVl^ ^ (2.55) we see that 2Im(Afc) / \af dS = - ^lim^ { / ( |fc|^ |ti;a|^ -h + Im{k) J [\kf\Wa\'' dWa dS dn + \\^Wa\^)dv\ ) DR (2.56) Now, if Im(AA:) > the conclusion a ~ on iS readily follows If Im(Afc) = and Im(A:) > we obtain Wa = in D^ Finally, if Im(Afc) = and Im(A:) = we get WxaR-^oo Js l^ol ^'^ "= ^' whence Wa = in Dg follows Application of the jump relations (2.50) finishes the proof of the theorem It is noted that the same strategy can be used for proving theorems 2.2 and 2.4 For instance, let Ua be the single-layer potential with density a € 1/^(5) satisfying Wa = in A Then lim ||tia(.4-M0)ll2,s = 0' (2.57) Um ^{.+hni.)) +a = 2,S With Sh = { y / y = X -(- hn{x), x € S, h > 0} being a parallel exterior surface we find that jua^dS = I uai.+hn{.))^{.+hn{.)) (2.58) SH X (1 - 2hH + h^K) dS for sufficiently small ft, whence < C\\Ua{.+hn{.))\Us ^(.-fftn(.)) (2.59) 2,S SH 32 CHAPTER II THE SCALAR HELMHOLTZ EQUATION follows Consequently, lim fua^dS h-^o^ J an =0 (2.60) and the conclusion follows as in theorem 2.6 T H E O R E M 2.7: Consider Di a bounded domain of class C^ with boundary S and exterior Dg, Let the combined potential Wa =^ Ua '\' Afa '^ith density a G i ^ ( ) and Im{Xk) > satisfy (2.61) Wa = in Dg Then a^O on S Proof: From the jump relations for the single- and double-layer potential we get lim \\wa{^-hn{.))-h Xa\\2 s = 0, (2.62) lim /i—0+ ||^(.-M.))-«| = Then, applying the first Green theorem in the region D^h bounded by the parallel interior surface 5_/i, S-h = { y / y = x - /in(x), x G 5, /i > 0} , letting /i —> 0-1- and using -A / \a\^ dS = lim / Wa^dS, J /1-.0+ J an (2.63) we find that - Im(Afe) / |a|^ d = Im(A:) / (|A:|^ \wa\'^ -f \Vwaf) S dV (2.64) Di Since Im(Afc) > and Im(fc) > we conclude that a ~ on GREEN'S FORMULAS AND SOLUTION ESTIMATES A basic tool in studying the boundary-values problem for acoustic scattering is provided by Green's formulas Consider Di a bounded domain of class C^ with boundary S and exterior £>«, and let n be the unit normal vector to S directed into D^ Let u € ^{Ds) be a radiating solution to the Helmholtz equation in Dg Then we have the Green formula (2.65) 33 GREEN'S FORMULAS AND SOLUTION ESTIMATES A similar result holds for solutions to the Helmholtz equation in bounded domains With u £ 3?(I>t) standing for a solution to the Helmholtz equation in Di we have -.%}=/h^tt?r-sw»'-.'' d5(y) / x€D ' X€ A (2.66) In the literature, Green's formulas are also known as the Helmholtz representations Next we will derive some estimates for the solutions to the Dirichlet and Neumann problems We begin with the exterior Dirichlet boundary-value problem The departure point is the associated boundary-value problem for the Green function G^ = G^(x,y), y e Ds, consisting in the Helmholtz equation AxGHx,y) + k^G'{x,y) = -6{x ~ y), x G D., (2.67) the boundary condition GHx,y) = , x € , (2.68) and the radiation condition ^ VxGHx,y) - jfcGnx,y) = o (J^A as |x| -^ oo, (2.69) uniformly for all directions x / |x| The superscript indicates that we are dealing with the Green function satisfying the Dirichlet boundary condition on S, Application of Green's second theorem in the domain Ds leads to dGHx,y) ^^(y) = r^siX ) dn{x) d5(x), y€Ds (2.70) We use the Cauchy-Schwartz inequality to obtain the estimate ll^«lloo,Gs = s u p |tx(y)| = y€Gs ju sup yeGs < dG\yi,y) (X) dn{x) dS(x) aGMx.y) d5(x) an(x) sup \ f\u.(x)fdS{x) = Ch« l l , ' \ (2.71) 34 CHAPTER II THE SCALAR HELMHOLTZ EQUATION where Gs is a closed subset of Dg and C = sup iI dG\x,y) d5(x) 9n(x) (2.72) In order to derive a similar estimate for the solution to the exterior Neumann problem we consider the Green function G^ = G^(x,y), y e £)«, satisfying the Helmholtz equation in Da, the Neumann boundary condition dG\-K,y) dn{x) 0, X e 5, (2.73) ^ ( x ) G ( x , y ) d ( x ) , y € £>,; (2.74) and the radiation condition We get us{y) = -J thus the estimate dug K I U G , < ^ dn (2.75) 2,5 holds in any closed subset Gg of Dg and for some constant C depending on and Gg Finally, for the exterior impedance boundary-value problem we consider the Green function G^ = G^(x,y), y e Dg^ satisfying the Helmholtz equation in JD^, the boundary condition G(x,y)-7 Q^^^^ -0,x€5, (2.76) and the radiation condition As before, application of Green's theorem in the domain Dg gives the representation My)=J r^W""'>'^w| ag'(x,y) dn{x) d5(x), y G Dg (2.77) Therefore, in any closed subset Gg of Dg the estimate dug I K I U G , t) yield u = in Di We note that the idea to employ the Fredholm alternative in two different dual systems for investigating the smoothness of a solution if the right side of the equation has a certain smoothness is due to Hahner [71] In the case of the Neumann problem we may employ the same arguments to show that if /i G L'^{S) is a solution of (2.91) then h € C^'"(5) Consequently, u given by (2.93) belongs to C^'"(jDi) and (2.91) gives u = in A provided that k i p{Di) NOTES AND COMMENTS Martin [99] showed that the null-field equations (2.82) and (2.86), are equivalent with some integral equations of the second kind, which possess an unique solution for all frequencies These integral equations are similar to (2.90) and (2.91), but they contain a new symmetric fundamental solution gi (x, y,A;) instead of ^(x, y,k) Note that gi (x, y,A;) differs from p(x, y,fc) by a finite linear combination of products of radiating spherical waves This equivalence allows Martin to conclude the unique solvability of the nullfield equations An approach similar to that given in Section was taken by Colton and Kress [33]  ... CHAPTER II THE SCALAR HELMHOLTZ EQUATION BOUNDARYVALUE PROBLEMS IN ACOUSTIC THEORY Acoustic waves are associated only with local motions of the particles of the fluid and not with bodily motion of the. .. boundary S and exterior Ds- In the scattering of time-harmonic acoustic waves by a sound-soft obstacle, the pressure of the total wave vanishes on the boundary This leads to the direct acoustic. .. obstacle scattering problem: given UQ as an entire solution to the Helmholtz equation representing an incident field, find the total field u = Ug -{ - UQ satisfying the Helmholtz equation in Dg and the